TITLE MARCH 2012 - Pakistan Academy of Sciences
TITLE MARCH 2012 - Pakistan Academy of Sciences
TITLE MARCH 2012 - Pakistan Academy of Sciences
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Proceedings <strong>of</strong> the <strong>Pakistan</strong> <strong>Academy</strong> <strong>of</strong> <strong>Sciences</strong> 49 (1): 1-8 (<strong>2012</strong>)<br />
Copyright © <strong>Pakistan</strong> <strong>Academy</strong> <strong>of</strong> <strong>Sciences</strong><br />
ISSN: 0377 - 2969<br />
<strong>Pakistan</strong> <strong>Academy</strong> <strong>of</strong> <strong>Sciences</strong><br />
Original Article<br />
On Regions <strong>of</strong> Variability <strong>of</strong> Some Differential Operators<br />
Implying Starlikeness<br />
Sukhwinder Singh Billing*<br />
Department <strong>of</strong> Applied <strong>Sciences</strong><br />
Baba Banda Singh Bahadur Engineering College<br />
Fatehgarh Sahib-140 407, Punjab, India<br />
Abstract: In this paper, we prove a subordination theorem and use it to extend the regions <strong>of</strong> variability <strong>of</strong><br />
some differential operators implying starlikeness <strong>of</strong> normalized analytic functions. Mathematica 7.0 is<br />
used to show the extended regions <strong>of</strong> the complex plane.<br />
Keywords: Analytic functions, Starlike functions, Differential subordination.<br />
2000 Mathematical Subject Classification: Primary 30C80, Secondary 30C45.<br />
1. INTRODUCTION AND<br />
PRELIMINARIES<br />
Let be the class <strong>of</strong> functions f , analytic in the<br />
open unit disk E { z:| z| 1} and normalized by<br />
the conditions f(0) f(0) 1 0 . Denote by<br />
* ( ), the class <strong>of</strong> starlike functions <strong>of</strong> order <br />
which is analytically defined as follows:<br />
<br />
* zf ()<br />
z <br />
<br />
( ) f<br />
: <br />
, zE, 0 1 .<br />
f()<br />
z <br />
<br />
*<br />
We write <br />
* (0) , the class <strong>of</strong> univalent<br />
starlike functions w.r.t. the origin. Obtaining<br />
different criteria for starlikeness <strong>of</strong> an analytic<br />
function has always been a subject <strong>of</strong> interest. A<br />
number <strong>of</strong> criteria for starlikeness <strong>of</strong> analytic<br />
functions have been developed. We state below<br />
some <strong>of</strong> them.<br />
Miller et al [4] studied the class <strong>of</strong> -convex<br />
functions and proved the following result.<br />
Theorem 1.1. If a function f satisfies the<br />
differential inequality<br />
zf ( z) zf ( z)<br />
<br />
(1 ) <br />
1 0, z E,<br />
f ( z) f ( z)<br />
<br />
where is any real number, then f is starlike in<br />
E.<br />
Later on, Fukui [1] proved the more general<br />
result given below for the class <strong>of</strong> -convex<br />
functions.<br />
Theorem 1.2. Let , 0 be a given real<br />
number. For all z E , let a function f <br />
satisfy<br />
zf ( z) zf ( z)<br />
<br />
(1 ) 1 <br />
f ( z) f ( z)<br />
<br />
<br />
,0 1/ 2,<br />
2(1 )<br />
<br />
(1 ) ,1/ 2 1.<br />
2<br />
Then<br />
f * ( )<br />
.<br />
Lewandowski et al [2] proved the following<br />
result.<br />
_____________________<br />
Received, February 2011; Accepted, March <strong>2012</strong><br />
*Email: ssbilling@gmail.com
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